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Full text of "Mathematical And Physical Papers - Iii"

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- I be the index of e in (13) when x is equal to one wave's length,
we have
^sin^r.0==Z,/Acos^p.a? = 27r, whence Z = 2?rtan^, <fz = 0-6172;
so that the intensity, supposed to vary as the square of the amplitude of vibration, would be diminished in the ratio of 2*625 to 1. Supposing the period of vibration to be the ^dth part of a second, which would correspond to a note of moderate pitch, and taking the velocity of propagation at 1100 feet per second, we should have 44 inches for the length of one wave. Hence in travelling 20 yards, or 16*36 wave-lengths, the intensity would be diminished in the ratio of (2'625)16'36 to 1, or about 7 millions to 1. A decrease of intensity like this is utterly contrary to observation, and therefore we are really compelled to suppose that the ratio of q to n is either very much greater or very much less than what has just been determined. Since in the case supposed n = 27TT-"1 = 600?r, we get from (16)
2 = 2198...........................(18),
which, it is to be remembered, is referred to a second as the unit of time.
Let us now, adopting this value of q, examine a little at what rate a small portion of heated air, situated in other air which has not been heated, would cool by radiation. If 9 be the excess of the temperature of the heated air over that of the surrounding air, we should have, supposing 6 to be sufficiently small to allow us to adopt Newton's law of cooling,
de 3**';
from which it follows that the excess of temperature would be diminished during the time t in the ratio of cfjt to 1. It would follow from the numerical value of q above given, that, even in so short a time as the hundredth part of a second, the temperature would be reduced in the ratio of about 3514 millions to 1. Such rapidity of cooling as this is utterly contrary to observation, Put a poker into the fire, and when it is hot look along it, and an ascending stream of heated air will be rendered visible by the distortion which it produces in objects seen through it, in consequence of the diminution of refractive power accompanying the rarefaction produced by heat. But were the rate of cooling